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New Meanings of the Fine Structure Constant

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New Meanings of the Fine Structure Constant New Meanings of the Fine Structure Constant Rodolfo A. Frino – January 2015 Electronics Engineer Degree from the National University of Mar del Plata Abstract In this paper I shall develop the simplest relativistic model of the hydrogen atom. On one hand, this model is incomplete since it neglects the spin of the electron. On the other hand, it incorporates relativistic effects which are not taken into account by the atomic models of Bohr and Schrodinger. In the light of this simple theory it would seem that the Rydberg constant, as far as the spectral lines is concerned, is just the result of the non-relativistic approach used by Bohr and not a true indication of the dependence of the spectral lines on this constant. This formulation emphasizes the significance of the fine-structure constant in the atomic structure of hydrogen and reveals two new meanings of this constant and its connection with life. Additionally this investigation introduces three new constants of Nature a) the minimum speed of light for the existence of hydrogen, c H , b) the energy-distance constant, ud , and c) the quantum energy-distance constant, uq . Keywords: Hydrogen atom, spectral lines, Lynman series, Balmer series, Paschen series, Planck's constant, Balmer formula, Rydberg formula, Rydberg constant, fine-structure constant, Sommerfeld's atomic theory. 1. Introduction Spectroscopic experiments carried out in the late 19th century showed that atoms emitted radiation at specific frequencies. Moreover, it was found that these frequencies were quantized. In 1885 Johann Balmer discovered an empirical formula describing the spectral lines emitted by hydrogen. This emission happens when an electron makes a transition between energy levels n and energy level 2. Balmer formula is 1 1 1 = R∞ − λ ( 22 n2 ) Where R∞ is the Rydberg constant. This constant is known with greatest accuracy (also referred as precision) and, according to CODATA 2010, its value is −1. R∞ =10 973 731.568 539(55)m In 1889, Johannes Rydberg generalized Balmer's formula by proposing another relationship to describe the hydrogen emission lines. This emission takes place when an electron makes a transition between energy levels ni and n f . His formula is 1 1 1 = R − λ ∞ 2 2 ( n f ni ) New Meanings of the Fine Structure Constant - v5. Copyright © 2011-2015 Rodolfo A. Frino. All rights reserved. 1 Rydberg formula becomes identical to Balmer's formula when n f equals 2. The explanation of the frequencies of the radiation emitted by the atom remained a mystery until 1913 when Niels Bohr introduced the first quantum hybrid theory of the hydrogen atom. This theory successfully explained not only the spectral lines predicted by Rydberg's formula but also the spectral lines of ionized Helium (and other hydrogenoid) and the size of the hydrogen atom. Bohr's idea was that when an electron makes a transition from a higher energy state ( ni ) to a lower energy state ( n f ) the excess energy must be given out as a photon of energy hf . Then, and according to Einstein, he was able to write the following relationship hf = Ei−E f Bohr also postulated the electron angular momentum was an integer multiple of ℏ . Mathematically this means that mv r = nℏ Where n is a quantum number (n=1, 2, 3, …). Furthermore, he demanded the electron did not lose energy in its orbit around the proton. In the present formulation we use the de Broglie condition which states that the circumference of the electron's orbit must be an integer multiple of the de Broglie wavelengths of the electron. Mathematically this can be expressed as follows 2π r = nλ Both postulates are equivalent since they are the same equation written in a slightly different way. The present formulation is relativistic in nature but unlike Sommerfeld's theory it only considers circular orbits and therefore uses only one quantum number. Thus this is the simplest relativistic atomic theory we can think of. This theory has almost all the limitations found in the Bohr theory, except that the fine-structure constant, α , emerges as a natural feature of the formulation. In Bohr's theory the fine-structure constant is “visible” only in the expression for the wavelengths of the spectral lines (see table 2) and, as far as I know, nobody realized about the existence of this constant until 1916 when Sommerfeld developed his atomic theory. The relativistic treatment of the formulation presented in this paper is what allows us to unveil the α -dependent formulas for the “allowed” atomic radii, the “allowed” energy levels and the “allowed” frequencies/wavelengths of the spectral emission lines of hydrogen. The significance of this theory is, therefore, based on the fact that this formulation is the simplest relativistic model of the hydrogen atom that is capable of predicting, explicitly, the fundamental relationship between the fine-structure constant and the atomic structure of hydrogen. According to this formulation the radius of the atom (as we shall see in section 2.3) depends on the fine-structure constant, as opposed to Bohr's theory. This is a profound conceptual difference between the two theories. 2. The Relativistic Model of the Hydrogen Atom In order to improve the structure, reading and understanding of this paper, the development of this theory is divided into the following subsections: New Meanings of the Fine Structure Constant - v5. Copyright © 2011-2015 Rodolfo A. Frino. All rights reserved. 2 2.1. Nomenclature and Equations 2.2. Electron Orbital Velocity (Allowed velocities) 2.3. Radii of the Orbits (Allowed radii) 2.4. Atomic Energy Levels (Allowed energy levels) 2.5. Frequencies of the Spectral Lines (Allowed spectral frequencies) 2.6. Wavelengths of the Spectral Lines (Allowed spectral wavelengths) 2.1. Nomenclature and Equations We shall use the following nomenclature for the constants and variables used in this paper Nomenclature n = principal quantum number i = another symbol for the principal quantum number n (used in frequency and wavelength formulas) j = another symbol for the principal quantum number n (used in frequency and wavelength formulas) h = Planck's constant ℏ = Planck's constant over 2π (reduced Planck's constant) k = proportionality constant of Coulomb law ϵ 0 = permittivity constant or permittivity of free space α = fine-structure constant (electromagnetic coupling constant) or alpha q1 = charge 1 (e.g. proton charge) q2 = charge 2 (e.g. electron charge) e = elementary charge m = relativistic mass of the electron m0 = electron rest mass p = relativistic momentum of the electron β = velocity ratio (electron orbital velocity over the speed of light in vacuum) λ = de Broglie wavelength of the electron λCe = Compton wavelength for the electron ƛCe = Compton wavelength for the electron over 2π (reduced Compton wavelength) λ s = wavelength of a given spectral line of hydrogen λij = wavelength of the spectral line corresponding to the electron transition between the energy levels i and j λij ( A)= wavelength of the spectral line corresponding to the transition between the energy levels i and j in Angstroms f = frequency of the spectral lines f ij = frequency of the spectral line corresponding to the electron transition between the energy levels i and j r = hydrogen atomic radius r n = hydrogen atomic radius r 1= hydrogen atomic radius (for n = 1) a0 = hydrogen atomic radius (for n = 1) (Standard name) v = orbital velocity of the electron vn = orbital velocity of the electron for level n New Meanings of the Fine Structure Constant - v5. Copyright © 2011-2015 Rodolfo A. Frino. All rights reserved. 3 v1 = orbital velocity of the electron for the ground level (n=1) c H = orbital velocity of the electron for the ground level (n=1) which is equal to the the minimum value of the speed of light necessary for the hydrogen atom to exist c = speed of light in vacuum F = attractive electrostatic force between the proton and the electron F C = centripetal force F JeV = conversion factor from Joules to electron-volts E rel = total relativistic energy of the hydrogen atom E = energy levels of the hydrogen atom E n = energy levels of the hydrogen atom E 1 = energy of the ground level of the hydrogen atom K = kinetic relativistic energy of the electron of a hydrogen atom K p = kinetic relativistic energy of the proton of a hydrogen atom (neglected) U = potential energy of the electron in a hydrogen atom ud = energy-distance constant (J-m) uq = quantum energy-distance constant (J-m) Equations The equations given below are the equations we shall use in the development of this theory (the are other equations, such as the Compton wavelength of the electron, that are not included here because they are not essential to the development of this theory). I have not included the equation for the fine-structure constant in this section as I assume that we do not know of its existence yet. This constant will emerge explicitly in the next sections. Thus, we shall assume that I) the circumference of the electron's orbit equals a whole number of wavelengths (pilot wavelength) of the electron. Mathematically this means that 2π r = nλ de Broglie standing wave condition (2.1-1) where n = 1, 2, 3, 4,... II) the wavelength and the relativistic momentum of the electron are related through the de Broglie’s relationship h λ = de Broglie equation (pilot wavelength) (2.1-2) p III) the momentum of the electron is p = mv Momentum (2.1-3) IV) the mass of the electron is given by the Einstein's relativistic mass New Meanings of the Fine Structure Constant - v5. Copyright © 2011-2015 Rodolfo A. Frino. All rights reserved. 4 m m = 0 Relativistic mass (2.1-4a) √1−β 2 v β = Velocity ratio (2.1-4b) c V) the electrostatic force between the proton and the electron is described by the Coulomb's law q q F = k 1 2 Coulomb's law (2.1-5a) r 2 1 k = Electric constant in terms of the permittivity constant (2.1-5b) 4π ϵ 0 VI) the centripetal force exerted on the electron is relativistic m v2 F = Relativistic centripetal force (2.1-6) C r VII) the potential energy of the electron at a distance r from the proton is given by the following law q q U =−k 1 2 Potential energy (2.1-7) r we shall assume that the potential energy depends on the spatial coordinates and does not depend on time.
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